# Tagged Questions

Galois theory allows one to reduce certain problems in field theory, especially those related to field extensions, to problems in group theory. For questions about field theory and not Galois theory, use the (field-theory) tag instead. For questions about abstractions of Galois theory, use (galois-...

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### Galois theory reference request

I ran into the following description of Galois theory in Gelfand and Manin's Methods of Homological Algebra. But this looks like nothing like the Galois theory I know that deals with subgroups and ...
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### Galois group of $x^{15}-1$

Let $\zeta$, $\eta$, $\omega$ denote the primitive fifteenth, fifth, and cube roots of unity. a) Describe all the automorphisms in $G=G(\mathbb Q (\zeta)/ \mathbb Q)$. b) Show that $G$ is isomorphic ...
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### Find the field of intersection

Let $\mathbb{F}_{p^{m}}$ and $\mathbb{F}_{p^{n}}$ be subfields of $\overline{Z}_p$ with $p^{m}$ and $p^{n}$ elements respectively. Find the field $\mathbb{F}_{p^{m}} \cap \mathbb{F}_{p^{n}}$. Could ...
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### Proving irreducibility of polynomial over various fields

I am working on a problem set from an online course and am struggling to understand a key concept related to proving reducibility / split-ability etc. in a finite field $\mathbb{F}_{p^n}$, as well as ...
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### Field algebraically closed

The problem is Let $E$ a finite extension of $F$ and $E$ is algebraically closed, show that $F$ is perfect. I know that a field $F$ is called perfect if every irreducible polynomial in $F[x]$ is ...
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### Finding the Galois Correspondence of polynomial $t^4-2$

For the polynomial $t^4-2$ in $\Bbb Q[t]$, the splitting field is given by $\Bbb Q(\alpha, i)$ where $\alpha$ is $2^{1/4}$. I figured out that the Galois group of this polynomial is the dihedral ...
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### Solvability of Artin-Schreier Polynomial

I'm having a hard time trying to prove that the polynomial f(x) = x^p - x - 1 in Z_p[x] is not solvable by radicals even though its Galois Group is solvable. So far, I have shown that the ...
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### Galois groups of quintics

I'm trying to determine which subgroups of $S_5$ occur as the Galois group of an irreducible quintic $f\in\Bbb{Z}[X]$. I know such a subgroup of $S_5$ should be transitive, leaving only five ...
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### Finding polynomial with Galois group $S_n$.

I'm studying the proof that for every $n\in \mathbb{N}$, there exists a polynomial $f\in \mathbb{Q}$ such that $\mbox{Gal}(E/\mathbb{Q})\cong S_n$, with $E$ the splitting field of $f$ over $\mathbb{Q}$...
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### Unramified Galois extension of a function field of a curve - definition

Reading some papers I've encountered the following: Let $X$ be a smooth curve over a field $K$ with function field $K(X)$. Consider the maximal Galois extension of $K(X)$ which is unramified over $X$. ...
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### Calculate the degree of $[\mathbb Q(a):\mathbb Q]$

How to determine $[\mathbb Q(a):\mathbb Q]$ for $a\in\overline{\mathbb Q}\setminus\mathbb Q$ with $a^p=1$ with $p$ prime? So $a$ is an algebraic, complex number which cannot be written as a quotient. ...
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### Galois field extension and number of intermediate fields

Given the galois extension $L/K$, where $L=\mathbb{Q}(\zeta, \sqrt[3]{2})$. I've already calculated that $[L:\mathbb{Q}]=6$. I'm trying to prove that the number of intermediate fields is also 6, ...
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### Calculating $Aut(\mathbb{Q}(\sqrt[3]{2}/\mathbb{Q}(\zeta))$

I am considering the Galois extension $L/\mathbb{Q}$, where $L = \mathbb{Q}(\sqrt[3]{2}, \zeta)$, and $\zeta$ is a primitive cube root of unity. I've found that $Aut(L/\mathbb{Q})$ can be viewed as ...
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### Let $K = \mathbb{Z_p(t)}$ be the field of rational functions and let $f(x) = x^p - x - t \in K[x]$ and let $E/K$ be the splitting field of $f(x)$.

Show $f(x)$ is not solvable by radicals and $Gal(E/K) \cong Z_p$. I was just reading Galois theory from the textbook "Galois Theory - Rotman" and I stumbled acrossed an exercise that looked ...
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### Intermediate field extensions and Degree of field extension

I was wondering whether there is a relation between $[L:K]$ and the number of intermediate fields $F$, where $K\subseteq F\subseteq L$. If there is then can someone please explain why. What if $L/K$ ...
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### Construct degree $n$ field extension with no intermediate field

I want to construct degree $n$ field extension with no intermediate field for each $n$. I know for any finite group $G$ there is a Galois extension $K/F$ so that $Gal(K/F)$ is $G$. So my idea was to ...
Let $G=S \times T$ where $S$,$T$ are both finite cyclic groups. Question 1: Is it true that there exists a Galois finite extension $L/F$ such that $Gal(L/F) \cong S \times T$? (I can't recall if ...